About subgroups rich in transvections

A subgroup H of the full linear group G=GL(n,R) of order n over the ring R is said to be rich in transvections if it contains elementary transvections tij(α)=e+αeij at all positions (i,j), i≠j (for some α∈R, α≠0). This work is devoted to some questions associated with subgroups rich in transvections. It is known that if a subgroup H contains a permutation matrix corresponding to a cycle of length n and an elementary transvection of position (i,j) such that (i-j) and n are mutually simple, then the subgroup H is rich in transvections. In this note, it is proved that the condition of mutual simplicity of (i-j) and n is essential. We show that for n=2k, the cycle π=(1 2 …n) and the elementary transvection t31(α), α≠0, the group ⟨(π),t31(α)⟩ generated by the elementary transvection t31(α) and the permutation matrix (cycle) (π) is not a subgroup rich in transvections.